Computation of Gross-Keating invariants

Abstract

The Gross-Keating invariant of a half-integral matrix over a p-adic integer ring is a fundamental concept in the study of quadratic forms, and has important applications for Siegel modular forms and arithmetic geometry. We introduce the Mathematica package computeGK, a computer program for calculating the Gross-Keating invariant and the Siegel series of a half-integral matrix over Z p , as well as other related quantities. As a by-product, we obtain a table of the arithmetic intersection numbers related to the classical modular polynomials using the explicit formula of Gross and Keating.